**1. Introduction**

Any method for the solution of crystal structures or for Rietveld refinement of such structures needs at least the set of symmetry matrices of the given space group (SG). In data bases and concise standard descriptions of crystal structures, the complete list of atoms is reduced to atoms representatives and a space group symbol. Since symmetry operations are tensor objects and their matrix form depend on the origin and the orientation of the coordinate system, the symbol must identify not only the *space group type* but also the *space group description*. For this reason standard space group descriptions called *conventional* were tabulated since the year 1935. Contemporary descriptions are based on the 3rd series of the *International Tables for Crystallography* [1], frequently referred to as ITA83. In a common practice one takes a space-group symbol, usually an international Hermann-Mauguin (H-M) symbol, as the index and obtains the symmetry operators from the printed tables. This approach guarantees the conventional descriptions of space groups, 'standard' ordering of symmetry matrices, its geometric interpretation, etc. but has a disadvantage of being limited to the tabulated representations of space groups. Moreover, huge amount of data and a possibility of human errors in transferring the symmetry information to applications involving crystal symmetry [2,3] leads to the generally accepted conclusion, that the automated derivation of space-group information becomes essential when this information is routinely required, especially in the case of higher symmetry. In the years 1960-1980 welldocumented algorithms [4,5] translating H-M symbols into a set of generators, which are then used to build a full set of symmetry matrices, were developed. Differences in the generated space group descriptions, caused by ambiguities in H-M symbols prompted in [6- 8], brought to procedures based on explicit-origin generators, that is on symmetry operations with specified complete translation vectors not only on its characteristic components – glide or screw vectors. The explicit-origin generators were organized into lengthy [7,8] or concise space-group symbols and used for the SG generation. The latter

© 2012 Stróż, licensee InTech. This is an open access chapter distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Stróż, licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 246 Recent Advances in Crystallography

symbol, known as the *Hall symbol*, was applied in [6,9]. It should be also noted, that according to the transformational concept [10], the one from equivalent sets of generators published in ITA83 can serve as the 'starting point' for any space group description in the ITA83-style.

The efficiency of space groups construction from the generators is also considered in the literature. Trivial approaches based on any symmetry operations, treated as the group generators, are redundant and completely ineffective. For example, the number of matrix multiplications and matrix comparisons needed for the generation of space group F*m*3�*m* has been estimated in [9] at 18528 and 1.7 million, respectively. Such huge number of mathematical operations can be drastically reduced at the expense of an algorithm complexity. In the approach described in reference [9], the improved algorithm was approximately 100 times faster than the simplistic method. The natural limit of such improvements corresponds to the non-redundant methods, that is procedures in which each matrix multiplication gives a new symmetry operation and thus matrix comparisons are not needed at all. Such method was based on representation of 32 crystallographic point groups in terms of cyclic groups and their products [7,8]. The 'composition series' method [11], successfully adapted to conventional space group descriptions in ITA83 [12], can also be modified to a non-redundant form [13]. This method is especially interesting, since it repeats the ITA83-type order of symmetry operations, assuming the published sets of generators are used. Basing on the composition series, all space group descriptions, that is conventional or not conventional, may be standardized [13-15].

For specific purposes justified in [8,9], the set of symmetry matrices should be complemented by the 'secondary symmetry information' which originates from the geometric characterizations of the symmetry operations given in ITA83 in the form of the operation symbols and the diagrams of symmetry elements. The symmetry operations are the foundation for the Hermann-Maugin symbols and to derive information important in understanding the Wyckoff positions, to find equivalent descriptions of crystal structures or to find transformations between different algebraic descriptions of the same space group.

The matrix-column pair of a space-group operator contains all the information that is needed for a geometrical characterization of the space-group operator and the corresponding symmetry element. Procedures for doing this are well documented in the literature [8,9,16-18]. Algorithms for the determination of (i) the type of rotation, (ii) the rotation angle φ given as n = 360/ φ, (iii) the sense of rotation, (iv) the orthogonal decomposition of the translation vector of the space group operator into its intrinsic and location-dependent components, (v) the orientation and the location of a geometric element, (vi) the inversion point on a geometric element in case of a rotoinversion symmetry operation, are given in the above references.

Even though the recipes are clear, the geometric meaning of symmetry matrices needs some additional conventions [18] to obtain a unique form, not necessarily presented in algorithms. Difficulties are connected with the positive direction of an axis line or plane traces specifications, or the selection of a point which fixes a geometric element in the space. Moreover, coding some glide vectors by the letters *a*, *b*, *c*, *g* causes that operation symbols are difficult to generate and are not generally understood, also in the community of crystallographers. Even though there are critical opinions like '*The symbols written under the heading Symmetry operations are actually descriptive symbols which repeat part of the information given by the diagram – actually they correspond to cosets representatives in the decomposition of the space group with respect to its translation subgroup….Solid state physicists would appreciate much more if these symbols are replaced by Seitz operators*' [19], the deriving of geometric descriptions, especially without any relations to the Crystallographic Tables, is crucial. It must be also remembered that symbols were used to give a scientific definition to a term '*symmetry element*' – the collective designation for a number of geometric concepts widely used by crystallographers, mineralogists and spectroscopists [20-22]. They were recommended by the *Ad-hoc committee on the nomenclature of symmetry* [21] for general use.

246 Recent Advances in Crystallography

not conventional, may be standardized [13-15].

operation, are given in the above references.

ITA83-style.

symbol, known as the *Hall symbol*, was applied in [6,9]. It should be also noted, that according to the transformational concept [10], the one from equivalent sets of generators published in ITA83 can serve as the 'starting point' for any space group description in the

The efficiency of space groups construction from the generators is also considered in the literature. Trivial approaches based on any symmetry operations, treated as the group generators, are redundant and completely ineffective. For example, the number of matrix multiplications and matrix comparisons needed for the generation of space group F*m*3�*m* has been estimated in [9] at 18528 and 1.7 million, respectively. Such huge number of mathematical operations can be drastically reduced at the expense of an algorithm complexity. In the approach described in reference [9], the improved algorithm was approximately 100 times faster than the simplistic method. The natural limit of such improvements corresponds to the non-redundant methods, that is procedures in which each matrix multiplication gives a new symmetry operation and thus matrix comparisons are not needed at all. Such method was based on representation of 32 crystallographic point groups in terms of cyclic groups and their products [7,8]. The 'composition series' method [11], successfully adapted to conventional space group descriptions in ITA83 [12], can also be modified to a non-redundant form [13]. This method is especially interesting, since it repeats the ITA83-type order of symmetry operations, assuming the published sets of generators are used. Basing on the composition series, all space group descriptions, that is conventional or

For specific purposes justified in [8,9], the set of symmetry matrices should be complemented by the 'secondary symmetry information' which originates from the geometric characterizations of the symmetry operations given in ITA83 in the form of the operation symbols and the diagrams of symmetry elements. The symmetry operations are the foundation for the Hermann-Maugin symbols and to derive information important in understanding the Wyckoff positions, to find equivalent descriptions of crystal structures or to find transformations between different algebraic descriptions of the same space group.

The matrix-column pair of a space-group operator contains all the information that is needed for a geometrical characterization of the space-group operator and the corresponding symmetry element. Procedures for doing this are well documented in the literature [8,9,16-18]. Algorithms for the determination of (i) the type of rotation, (ii) the rotation angle φ given as n = 360/ φ, (iii) the sense of rotation, (iv) the orthogonal decomposition of the translation vector of the space group operator into its intrinsic and location-dependent components, (v) the orientation and the location of a geometric element, (vi) the inversion point on a geometric element in case of a rotoinversion symmetry

Even though the recipes are clear, the geometric meaning of symmetry matrices needs some additional conventions [18] to obtain a unique form, not necessarily presented in algorithms. Difficulties are connected with the positive direction of an axis line or plane traces specifications, or the selection of a point which fixes a geometric element in the space. The analysis of symbol derivations as well as the contents of geometric features in the symbols, given in [15, 18, 23], suggests the possibility of defining new symbols, easier to calculate, universal and richer in geometric meaning.

Taking into account all above remarks, the most important features for the computer generation of space groups and their geometric characterization are: (i) a possibility of unique naming of any SG descriptions, (ii) an easy interpretation of SG symbols, that is a derivation of generators, (iii) effectiveness (non-redundancy) of group generation, (iv) reduced amount of predefined data, (v) full consistency with ITA83, (vi) an effective geometric characterization of obtained set of symmetry matrices. The generators selected by H. Wondratschek and used for the conventional space group descriptions in ITA83 meet some demands from the above list, but the geometric symbols need some modifications.

In this chapter a complete, effective and unique approach to the automated derivation of space-group information for a conventional as well as for a non-conventional space group description is proposed.
